How many values of $x$ with $0^\circ \le x < 360^\circ$ satisfy $\sin x = -0.73$?
Explanation: [asy]
pair A,C,P,O,D;
draw((0,-1.2)--(0,1.2),p=black+1.2bp,Arrows(0.15cm));
draw((-1.2,0)--(1.2,0),p=black+1.2bp,Arrows(0.15cm));
A = (1,0);
O= (0,0);
label("$x$",(1.2,0),SE);
label("$y$",(0,1.2),NE);

P = rotate(150)*A;
D = foot(P,A,-A);
draw(Circle(O,1));
label("$O$",O,SE);

draw((-1,-0.73)--(1,-0.73),red);
[/asy]

For each point on the unit circle with $y$-coordinate equal to $-0.73$, there is a corresponding angle whose sine is $-0.73$.  There are two such points; these are the intersections of the unit circle and the line $y=-0.73$, shown in red above.  Therefore, there are $\boxed{2}$ values of $x$ with $0^\circ \le x < 360^\circ$ such that $\sin x = -0.73$.